LECTURE 15 PROOF OF THE EXISTENCE OF WALRASIAN EQUILIBRIUM

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1 LECTURE 15 PROOF OF THE EXISTENCE OF WALRASIAN EQUILIBRIUM JACOB T. SCHWARTZ EDITED BY KENNETH R. DRIESSEL Abstract. We prove the existence of Walrasian equilibrium in a case of single labor sector and in a case of nonhomogeneous labor force. 1. Case of a Single Labor Sector In the previous lecture we derived a set of conditions necessarily satisfied at any price equilibrium. In the present lecture, we shall show that these conditions are also sufficient, and that they may all be satisfied simultaneously. In this way we shall establish the existence of a price equilibrium for the model introduced in Lecture 13, i.e., we shall show the existence of prices p such that the following conditions are satisfied (a) p 0, = 0,, n; moreover, for some k, p k > 0 (b) a (k) i (δ i π i )p = max subect to (k) a i φ i p q (k) p. (See Lecture 13 for a more precise statement.) (c) u (ν) (s (ν) ) = max subect to p s (ν) a (k) i (δ i π i )p h νk. k i (See Lecture 13 for a more precise statement.) 2010 Mathematics Subect Classification. Primary 91B50; Secondary 91B24. Key words and phrases. Walrasian Equilibrium, Price Theory. 1

2 2 JACOB T. SCHWARTZ (d) a (k) i (δ i π i ) s (l) 0, for all and i,k l a (k) i (δ i π i ) + s (k) i = 0 for all for which p > 0. i k k We recall that these conditions have the following briefly stated heuristic significance: (a) the price of each commodity is nonnegative; (b) each firm operates so as to maximize its profits subect to its capital limitations; (c) each consumption unit optimizes its own utility function subect to its own budgetary constraints, and (d) whatever is consumed must be produced, and overproduction cannot occur for a commodity whose price is positive. Having established the existence of a set of prices, production schemes, and consumption schemes satisfying (a) (d) above, we shall formulate a more general model, in which not one but several kinds of labor will be included, and show the existence of a general price equilibrium in that case also. 2. A Preliminary Existence Theorem We shall proceed by setting up an ansatz giving prices, production, consumption, etc., as functions of a rate-of-profit parameter ρ, and then choosing ρ so as to obtain a price equilibrium. In determining the form of our ansatz, we will of course be guided by the analysis carried out in the preceding lecture. Let us begin by assuming prices p (ρ) as solutions of the equation (15.1) (δ i π i )p (ρ) = ρ φ i p (ρ); i = 1,, n where ρ is a parameter; we know from Lecture 3 that a nonnegative (and nontrivial) solution of these equations exists for all values of ρ satisfying 0 ρ ρ max and that these prices satisfy condition (a), provided only that Π is a connected matrix. The matrix Π will be assumed to be connected throughout the present lecture. Note that this implies that the prices p (ρ), = 1,, n are strictly positive for

3 15. PROOF OF EXISTENCE OF WALRASIAN EQUILIBRIUM 3 all ρ, 0 ρ ρ max. The price p 0 (ρ) is positive if and only if ρ ρ max while p 0 (ρ max ) = 0. Next we choose the consumption levels s (ν) so as to maximize each of the utility functions u (ν) subect to the constraint (15.2) p (ρ)s (ν) = (ν) (ρ) where, as in the preceding lecture, we put (15.3) (ν) (ρ) = ρ,k q (k) p (ρ)h νk. As we saw in the preceding lecture, this condition determines the quantities s (ν) as functions of the rate of profit ρ. We define the total consumption s i (ρ) of the ith commodity as (15.4) s i (ρ) = f ν=1 s (ν) i (ρ). Finally, we determine the total production a i = m k=1 a(k) i of the ith commodity by the equation (15.5) a i (ρ)(δ i π i ) = s (ρ), = 1,, n. i=1 We put q = m k=1 q(k), = 1,, n for convenience. The following lemma then reduces the problem of establishing the existence of a price equilibrium to a more tractable form. Lemma Let the prices, consumption, and total production be determined from ρ as above. Then there will exist a determination of the production levels a (k) for the various firms satisfying m (15.6) = a (ρ) k=1 a (k) which, together with the prices and consumption schemes determined above, will satisfy the conditions (a) (d), i.e., will define a price equilibrium, provided that

4 4 JACOB T. SCHWARTZ (1 ) either a i (ρ)φ i p (ρ) = q p (ρ) i, or ρ = 0 and a i (0)φ i p (0) q p (0) and provided also that (2 ) either or ρ = ρ max and i, m a i (ρ)π i0 = s 0 (ρ) i=1 a i (ρ)π i0 s 0 (ρ). i=1 Proof: If the prices satisfy (15.1), they are nonnegative and not all zero so that (a) is satisfied. If in condition (1 ) we have ρ > 0, it is plain that we may divide the total production levels a (ρ) into individual-firm contributions a (k) as in (15.5), in such a way that a (k) φ i p i (ρ) = q (k) p (ρ) i, for each k, k = 1,, m. Similarly, if in condition (1 ) we have ρ = 0, we may divide the total levels into individual-firm contributions a (k) in such a way that a (k) φ i p i (0) q (k) p (0) i, for each k, k = 1,, m. Thus, if condition (1 ) is satisfied, we may in any case divide the total production levels a (ρ) into individual-firm contributions in such a way that no firm s production plans will require the outlay of more capital than it possesses; moreover, unless ρ = 0, each firm will make use of the whole of its capital. As we have seen in the previous lecture, this, coupled with (15.1), is ust the condition that each firm s profits be maximized. Thus (b) of our conditions, defining a price equilibrium, is satisfied. We have determined the consumption schemes s (ν) i (ρ) in such a way that (c) is automatically satisfied. Finally, since all the relations in (d) except the first, i.e., the = 0 relations, are satisfied in consequence of the way the a (k) i have been defined, and since condition (2 ) is the condition that the = 0 relation be satisfied also (recall that ρ = ρ max means that the price of labor is zero), condition (d) is also satisfied. Q.E.D.

5 15. PROOF OF EXISTENCE OF WALRASIAN EQUILIBRIUM 5 In the trivial case ρ max = 0, we have p 0 = 0 and ρ = 0, so that s (ν) = 0 for 1 by (15.2) and (15.3); thus s = 0 for 1 by (15.4) and a = 0 by (15.5), so that the above lemma tells us that a price equilibrium exists and is to be determined by our formulas (15.1) (15.5). We consequently may, and shall unless the contrary is explicitly indicated, assume in what follows that ρ max > 0. Let us define the demand for labor d 0 (p) by (15.7) d 0 (ρ) i a i (ρ)π i0 and recall that we have already defined the supply of labor s 0 (ρ) by (15.4). The following useful identity reduces the conditions of Lemma 1 to a more manageable form. Lemma Let a i (ρ), p i (ρ), s 0 (ρ), and d 0 (ρ) be as above. Then ( ) ( s 0 (ρ) d 0 (ρ))p 0 (ρ) = ρ a i (ρ)φ i p (ρ) q p (ρ). =1 =1 Proof: Consider the expression ( ) (15.8) a i (ρ)(δ i π i )p (ρ) s (ρ)p (ρ). i=1 All terms in this expression, except possibly the = 0 term, are zero by (15.5). Therefore (15.8) is ust a i (ρ)π i0 p 0 (ρ) s 0 (ρ)p 0 (ρ) i=1 or, equivalently, ( s 0 (ρ) d 0 (ρ))p 0 (ρ). But (15.8) can be written (using (15.1), (15.2), and (15.3)) as (15.9) ρ a i (ρ)φ i p (ρ) ρ q p (ρ). i=1 Therefore (15.10) ( s 0 (ρ) d 0 (ρ))p 0 (ρ) = ρ[ ai (ρ)φ i p (ρ) ] q p (ρ).

6 6 JACOB T. SCHWARTZ Q.E.D. Putting together Lemmas 1 and 2 yields the following corollary: Corollary Let prices and total production be determined from ρ as above. Then there will exist a determination of the production levels for the various firms satisfying (15.6), which, together with the prices and consumption schemes determined above, will define a price equilibrium, provided that either (a) 0 < ρ < ρ max and d 0 (ρ) + s 0 (ρ) = 0 or (b) ρ = ρ max and d 0 (ρ) + s 0 (ρ) 0 or (c) ρ = 0 and a i (0)φ i p (0) i, q p (0). Lemma 2 also yields trivially the following corollary, which states a fact used in the construction of the diagrams of the previous lecture and also gives an equivalent alternate version of the condition under (c) of the preceding corollary. Corollary (i) d 0 (0) + s 0 (0) = 0 (ii) (d/dρ)(d 0 (ρ) + s 0 (ρ)) ρ=0 = q p (0) i, a i (0)φ i (0)p (0). Proof: Statement (i) follows from Lemma 2, and statement (ii) follows from statement (i) and Lemma 2. Q.E.D. We have supposed that ρ max > 0. Using what we have proved we may distinguish three possibilities for s 0 (ρ) + d 0 (ρ). The first is that for some ρ, 0 < ρ ρ max, supply and demand for labor are equal, i.e., s 0 (ρ) + d 0 (ρ) = 0. The second is that for all ρ, 0 < ρ ρ max, s 0 (ρ) < d 0 (ρ). The third possibility is that for all ρ, 0 < ρ ρ max, it is the case that s 0 (ρ) > d 0 (ρ). Note that in any case, by Corollary 4, s 0 (0) + d 0 (0) = 0. In the first case, condition (a) of Corollary 3 is satisfied. In the second case, since s 0 (ρ) + d 0 (ρ) is positive for ρ > 0 and zero for ρ = 0, the derivative of s 0 (ρ) + d 0 (ρ) is nonnegative at ρ = 0, so that by Corollary 4, condition (c) of Corollary 3 is satisfied. If s 0 (ρ) + d 0 (ρ) is negative for ρ > 0, the condition (b) of Corollary 3 is satisfied. This establishes the existence of a price equilibrium in every case.

7 15. PROOF OF EXISTENCE OF WALRASIAN EQUILIBRIUM 7 The following theorem summarizes the necessary conditions for equilibrium derived in the preceding lecture and the sufficient conditions derived in the present section. Theorem Let φ 0 (ρ) = s 0 (ρ) + d 0 (ρ), where s 0 (ρ), d 0 (ρ) are defined as above. Then price equilibria correspond to those values 0 < ρ < ρ max for which φ 0 (ρ) = 0; and also to the value ρ = ρ max if and only if (d/dρ)[φ 0 (ρ)] ρ=0 0; and also to the value ρ = ρ max if and only if φ 0 (ρ max ) 0. At each such equilibrium the prices are unique; the total activity levels are also unique, and these totals may be divided among the firms in any way consistent with the requirement a (k) π i p i (ρ) q (k) p (ρ), k = 1,, m. i, 3. Price Equilibrium for a Nonhomogeneous Labor Force We now extend our price-equilibrium model to include several labor sectors, considered as originating labor commodities C 0, C 1,, C L. If ρ max = 0, no wages can be paid; hence the distinction between labor sectors is entirely irrelevant, and we return to the model of the preceding section. generality that ρ max > 0. Hence we may and shall assume without loss of In the present more general context, we shall reestablish results much like those of the two preceding lectures. We will find, in particular, that price equilibria exist whenever supply and demand for labor are simultaneously in balance in all of the labor sectors. As we have seen in Lecture 3, prices may be determined by the basic relation (15.1) (the range of summation of being extended from L to n) as functions of ρ and of the ratios θ ( = 0,, L) of the wages, where we normalize these ratios by demanding that L θ = 1. Thus, letting p (ρ, θ) be the prices so determined, we can write (15.11) (δ i π i )p (ρ, θ) = ρ φ i p (ρ, θ). = L = L

8 8 JACOB T. SCHWARTZ We shall assume throughout the present section that each form of labor is required as input to the production of at least one commodity, and shall also assume that the n n matrix Π, obtained by deleting all elements of π i but those with both i and between 1 and n, is connected. To construct a price equilibrium, we imitate the procedure of the preceding sections, as follows. The consumption levels s (ν) are again chosen so as to maximize (15.12) u (ν) (s ) over all s such that (15.13) = L p (ρ, θ)s = ρ i,k q (k) i p i (ρ, θ)h νk. This determines total consumption, defined by (15.14) s i (ρ, θ) ν s (ν) i (ρ, θ). Total activity levels are then defined by the equations (15.15) a i (ρ, θ)(δ i π i ) = s (ρ, θ), = 1,, n, i=1 and we define the demand for labor in the various sectors by (15.16) d (ρ, θ) = a i (ρ, θ)π i,, = 0,, L. i=0 The above formulae define individual-family consumption schemes, prices, and total production as functions of ρ and θ, but not the division of total production among the individual firms. It will be convenient in what follows to speak of those values of ρ and θ for which the total production levels a may be divided into the sum of individual-firm contributions a (k) to yield a set of individual-firm production levels, family consumption and labor schemes, and prices satisfying the conditions (a) (d) defining a price equilibrium as values ρ, θ corresponding to a price equilibrium. This terminology will be employed in stating a

9 15. PROOF OF EXISTENCE OF WALRASIAN EQUILIBRIUM 9 number of the theorems and lemmas which are to follow. Put φ (ρ, θ) = d (ρ, θ) + s (ρ, θ). We aim to obtain the following result. Theorem A. A price equilibrium will correspond to each of those and only those points 0 < ρ ρ max, θ 0,, θ L for which φ (ρ, θ) 0 for all = 0,, L, with equality whenever p (ρ, θ) 0; and also to those and only those points ρ = 0, θ 0,, θ L for which φ (0, θ) is nonnegative for = 0,, L and [ L ] (15.17) d/dρ p (ρ, θ)φ (ρ, θ) 0. ρ=0 B. Furthermore, there always exists at least one equilibrium point. The remainder of this lecture will be devoted to proving the above theorem. The proof of part A follows the development in Section 2 of this lecture quite closely, but to prove part B we must employ some deeper topological theorems, since we cannot make use of the basically one-dimensional division into three collectively exhaustive cases φ > 0, φ = 0, φ < 0 as in the last part of Section 2. Since equation (15.11) is homogeneous in the prices p, it only determines these prices up to a scalar factor. In the arguments which are to follow, it is important to make this factor definite and convenient to choose it in an appropriate way. For this reason, we preface our proof by a lemma characterizing a manner of normalizing the solutions p (ρ, θ) of (15.11), to which normalization we shall adhere in the remainder of the present section. Lemma The solution p (ρ, θ) of equations (15.11) may be made definite by imposing the additional requirement L (15.18) ρ + p (ρ, θ) ρ max.

10 10 JACOB T. SCHWARTZ The solutions determined in this way depend continuously on ρ and θ; moreover, they satisfy the condition (15.19) p (ρ, θ) > 0, = 1,, n. Proof: The factors θ have been chosen to satisfy L θ = 1. Thus, a solution of (15.11) will satisfy (15.18) if and only if p (ρ, θ) = (ρ max ρ)θ ; hence what we must do is prove the continuity and positivity of the solutions p (ρ, θ) of the equation (15.20) L L (δ i (1 + ρ)π i )p (ρ, θ) = (1 + ρ)(ρ max ρ) π i, θ ; =1 i = 1,, n. If we let v(ρ, θ) be the vector whose ith. component is L (1 + ρ) π i, θ and let p 1 (ρ, θ) be the vector of length n whose components are p (ρ, θ), = 1,, n, we may write the solution of (15.20) in vectorial form as (15.21) p 1 (ρ, θ) = (ρ max ρ)(i (1 + ρ) Π) 1 v(ρ, θ). This formula makes it evident that the proof of the present lemma will result immediately once we establish the following lemma. Lemma Let A be a connected positive matrix with dom(a) < 1, and let α be determined by the equation dom((1 + α)a) = 1, i.e., α = (dom(a)) 1 1. Then the matrix (15.22) (α ρ)(i (1 + ρ)a) 1 = J(ρ) is continuous and strictly positive in the whole closed interval 0 ρ α. Proof: The correctness of our statement in the interval 0 ρ < α follows at once from Lemmas 3.3, 3.5, and 3.6. Thus we have only to show that the matrix (15.22) has a limit as ρ approaches α from below, and that this limiting matrix is positive.

11 15. PROOF OF EXISTENCE OF WALRASIAN EQUILIBRIUM 11 Let v be the dominant eigenvector of A, so that (1 + α)av = v. Then (15.23) (α ρ)(i (1 + ρ)a) 1 v = {(α + ρ)/1 [(1 + ρ)/(1 + α)]}v = (1 + α)v, so that J(ρ)v is bounded. Since every component of v is positive, and since all the entries of J(ρ) are positive for ρ < α, it follows at once that J(ρ) remains bounded as ρ α. Thus we may extract a convergent subsequence from the family of matrices J(ρ); i.e., we may find a sequence ρ n α such that the sequence of matrices has a limit: lim n J(ρ n ) = J. Since the limit of a sequence of nonnegative matrices is nonnegative, it follows from (15.23) that J is nonnegative and Jv = (1 + α)v. Since (I (1 + ρ)a)j(ρ) = (α ρ), it follows on letting ρ approach α that (1 + α)aj = J; thus every column of the matrix J must be proportional to the column vector v; similarly, we may show that every row of the matrix J must be proportional to the uniquely defined adoint vector u which satisfies u(1 + α)a = u. Hence the elements J ik of J must have the form J ik = cv i u k. Since Jv = (1 + α)v, we must have cu v = 1 + α, so that c = (u v) 1 (1 + α) is positive, and thus J is positive and uniquely defined. Let ρ n be any other sequence of numbers approaching α from below for which lim n J(ρ n) = J exists. Then, by the argument of the preceding paragraph, we must have J = J. This shows that lim ρ α J(ρ) = J exists, establishing the continuity and positivity of J(ρ) in the closed interval 0 ρ α and completing the proof of the present lemma and of Lemma 7. Q.E.D. Next we give a lemma generalizing the key Lemma 2 of the preceding section. Lemma Let φ (ρ, θ) = d (ρ, θ) s (ρ, θ), = 0,, L as above, and put (15.24) φ L 1 (ρ, θ) = q p (ρ, θ) i, a i (ρ, θ)π i p (ρ, θ),

12 12 JACOB T. SCHWARTZ where q = q (k). Moreover, put (15.25) θ = [1 (ρ/ρ max )]θ, i = 0,, L and (15.26) θ L+1 = ρ/ρ max. Then (15.27) Moreover, L+1 φ (ρ, θ)θ = 0. (15.28) θ 0 and L+1 θ = 1. Proof: (15.29) By (15.11), (15.13), (15.14), and (15.15) we have [ ρ a i (ρ, θ)φ i p (ρ, θ) ] q p (ρ, θ) i, = a i (ρ, θ)(δ i π i )p (ρ, θ) s (ρ, θ)p (ρ, θ) i, = = = s (ρ, θ)p (ρ, θ) 0 a i (ρ, θ)π i p (ρ, θ) i = L 0 s (ρ, θ)p (ρ, θ) s (ρ, θ)p (ρ, θ) =1 =1 = L L ( s (ρ, θ) d (ρ, θ))p (ρ, θ) L φ (ρ, θ)p (ρ, θ). Our lemma follows immediately on using the definitions (15.25) and (15.26). Q.E.D. We may now conveniently make use of a result from elementary topology.

13 15. PROOF OF EXISTENCE OF WALRASIAN EQUILIBRIUM 13 To state this result, we have first to make a number of simple geometric definitions. Definition. The k-dimensional simplex is the subset σ of Euclidean k + 1-dimensional space defined by the formula { } k+1 σ = x x 0, x i = 1. Definition. The vertices of the k-dimensional simplex σ are the k + 1 points p () defined by i=1 (15.30) p () i = δ i, i, = 1,, k + 1. Definition. Let p ( 1),, p (r) be a set of r distinct vertices of the simplex σ. Then the side determined by these vertices is the subset σ 0 of σ defined by the formula (15.31) σ 0 = {x σ x = 0 unless = 1 or = 2 or or = r }. In terms of these definitions we may state the following useful topological theorem. Theorem Suppose an n 1-dimensional simplex is covered by n closed sets Σ 1,, Σ n, in such a way that for each r the r 1- dimensional side determined by the vertices p (i1),, p (ir) is contained in the union of the r sets Σ i1,, Σ ir. Then the sets Σ have a common point, i.e., Σ 1 Σ 2 Σ n. (For an elementary proof, see L. Graves, Theory of Functions of Real Variables (1946 Edn., p. 148).) We can use this theorem to establish the following. Lemma Let σ be the simplex defined by { } k σ = x x = 1; x i 0. =1

14 14 JACOB T. SCHWARTZ Let f (x 1,, x k ), = 1,, k be continuous functions defined on σ such that k x f (x 1,, x n ) = 0, x σ. =1 Then there is a point x 0 1,, x 0 n such that for every f (x 0 1,, x 0 n) 0, with equality wherever x 0 0. Proof: For each = 1,, k define the closed set Σ by writing (15.32) Σ = {x f (x) 0}. We first show that the r 1-dimensional side σ of σ with vertices p (i1),, p (ir) is contained in the union of the sets Σ i1,, Σ ir. After permuting the variables x, we may assume without loss of generality that i 1,, i r = 1,, r. Our side is then the subset of (15.32) defined by the equations x r+1 = x r+2 = = x k = 0 so that k x f (x) = 0 =1 everywhere along the side σ. Therefore for each x in σ, at least one function f must be nonnegative. Thus the side σ is contained in Σ 1 Σ r. By Theorem 10, then, the intersection of all the sets Σ is not void. Any point in the intersection is a point (x 0 1,, x 0 n) satisfying the conditions of the present lemma. Q.E.D. We may use Lemma 9 and Theorem 10 to derive the following lemma, which practically contains the main Theorem 6 which we are in the course of proving. Lemma There exists a point (ρ, θ) such that (15.33) φ (ρ, θ) 0, = 0,, L + 1.

15 15. PROOF OF EXISTENCE OF WALRASIAN EQUILIBRIUM 15 At this point we have φ (ρ, θ) = 0 if L + 1 and p (ρ, θ) 0 while φ L 1 (ρ, θ) = 0 if ρ 0. Proof: Let θ i, i = 0,, L + 1 be the quantities defined in the statement of Lemma 9. Since ρ may be varied arbitrarily between 0 and ρ max, and since the quantities θ may be varied arbitrarily subect only to the restrictions θ 0 and L θ = 1, it is plain that by choosing ρ and θ, the quantities θ may be varied arbitrarily subect only to the restrictions (15.28). It is clear from (15.25) and (15.26) that as long as θ L+1 1, i.e., as long as ρ ρ max, we may write ρ and θ as continuous functions of θ : (15.34) ρ = ρ(θ ), θ = θ(θ ). Thus the functions φ (ρ(θ ), θ(θ )) depend continuously on θ except at the vertex θ L+1 = 1. However, since (cf. Lemma 15.7) p (ρ, θ) = θ, even in the vicinity of ρ = ρ max, it follows from (15.11), (15.12) (15.14), (15.15), (15.16) and the formula following (15.16), and (15.24) (15.26) that φ (ρ(θ ), θ(θ )) depends continuously on θ even near θ L+1 = 1. Thus, by Lemma 11, there exists a ρ and θ such that φ (ρ, θ) 0, = 0,, L + 1, with equality if θ 0. Using (15.25) and (15.26), the present lemma follows at once. Lemma A point ρ = 0 satisfies (15.35) φ L 1 (0, θ) 0 if and only if (15.36) (d/dρ) Proof: (15.37) L φ (ρ, θ)p (ρ, θ) Use the identity ρ=0 0. L φ (ρ, θ)p (ρ, θ) + φ L 1 (ρ, θ) = 0 given by Lemma 9; differentiate and put ρ = 0. Now we shall prove part A of Theorem 6. Q.E.D. Q.E.D.

16 16 JACOB T. SCHWARTZ Proof of Part A of Theorem 6. Suppose first that ρ > 0. Let ρ, θ be a rate of profit and a set of wage ratios such that φ (ρ, θ) 0, 0 L+1 with equality whenever θ 0. It is clear from Lemma 10 that not all the prices p (ρ, θ) are zero; thus condition (a) is satisfied. Since the consumption schemes s (ν) (ρ, θ) are determined by (15.12) (15.13), they satisfy condition (c). Since the total activity levels a i (ρ, θ) are defined by (15.15) and since by hypothesis we have φ (ρ, θ) 0 with equality whenever p (ρ, θ) 0, we see that condition (d) is also satisfied. It only remains to show that condition (b) is satisfied; since with prices giving a uniform positive rate of profit a firm makes maximum profits if and only if it employs all its capital it is sufficient to show that the total activity levels a i (ρ, θ) can be divided into the sum of individual-firm contributions a (k) i in such a way that (15.38) = a i (ρ, θ), and (15.39) k,i, a (k) i k a (k) i φ i p (ρ, θ) = q (k) p (ρ, θ). It is plain that (15.38) and (15.39) can be satisfied simultaneously if and only if (15.40) a i (ρ, θ)φ i p (ρ, θ) = q p (ρ, θ), i, where q = k q(k). Formula (15.24) shows that (15.40) holds, since, by hypothesis, φ (ρ, θ) = 0 unless p (ρ, θ) = 0. Conversely, suppose that we have a price equilibrium corresponding to the rate of profit ρ > 0 and the wage ratios θ. We know that condition (b) implies that the equilibrium prices are (proportional to) p (ρ, θ), = L,, n. Since we have seen that all the prices p (ρ, θ), = 1,, n are then strictly positive, it follows from condition (d) that the total activity levels a i = k a(k) i, i = 1,, n must satisfy (15.15). Moreover, by condition (b), each firm must employ its total capital; thus, the dividend income of the νth family is necessarily given

17 15. PROOF OF EXISTENCE OF WALRASIAN EQUILIBRIUM 17 by (15.41) ρ i,k q (k) i p i h νk, which by condition (c) implies that the family s labor-consumption scheme must satisfy (15.12) (15.13). Next we observe that condition (d) implies that at equilibrium we have φ (ρ, θ) 0, with equality if p (ρ, θ) 0. Thus part A of Theorem 6 is proved in case ρ > 0. If ρ = 0, we may argue in almost the same way, making use however of the information given by Lemma 13. We have only to amend the sixth sentence of the first paragraph of the present proof to state that if ρ = 0, a firm makes no profit in any case, so that condition (b) degenerates by the further argument of the present proof to the condition (15.42) a i (0, θ)φ i p (0, θ) q p (0, θ), i, i.e., to the condition φ L 1 (0, θ) 0. The converse argument in case ρ = 0 is again much like the converse argument in case ρ > 0; we have only to amend the preceding paragraph with the remark that if profits are all zero, dividend income for each family is zero also. Thus part A of Theorem 6 is fully proved. Proof of Part B of Theorem 6: This follows at once from Part A and from Lemmas 12 and 13. Q.E.D. 4. A Summary Remark The analysis presented in the present lecture shows the Walrasian theory to be quite satisfactory from the purely mathematical point of view. On the other hand, we have seen above that its mathematical formulations are based upon unrealistically drastic assumptions, which artificially assume away all the Keynesian phenomena studied in Lectures In order to incorporate the neoclassical ideas of the three preceding lectures into a less hopelessly unrealistic model, it is necessary for us to construct a generalized equilibrium model combining neoclassical and Keynesian features. We now turn to this task.